Question

The wave number of the shortest wavelength of absorption spectrum of hydrogen atom is_ _ _ _
(Rydberg constant = \(109700\,\mathrm{cm^{-1}}\)).

Hint:

In absorption spectrum, shortest wavelength corresponds to series limit (highest transition).

Answer 1

Correct Answer: 27425

Step 1: Understanding the Concept:
In the absorption spectrum of Hydrogen, the shortest wavelength corresponds to the transition with the highest energy. This occurs when an electron at the ground state (\(n = 1\)) is excited to infinity (\(n = \infty\)), effectively ionizing the atom.
Step 2: Key Formula or Approach:
Wave number (\(\bar{\nu}\)) formula:
\[ \bar{\nu} = R \left( \frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}} \right) \]
: Detailed Explanation:
For shortest wavelength (maximum energy) in absorption:
\(n_{1} = 1\) (starting state)
\(n_{2} = \infty\) (final state)
\[ \bar{\nu} = R \left( \frac{1}{1^{2}} - \frac{1}{\infty^{2}} \right) \]
Since \(\frac{1}{\infty} = 0\):
\[ \bar{\nu} = R (1 - 0) = R \]
Given \(R = 109700\text{ cm}^{-1}\).
\[ \bar{\nu} = 109700\text{ cm}^{-1} \]
Step 3: Final Answer:
The wave number is \(109700\text{ cm}^{-1}\).